In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this clas...
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In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classification the lines lying on the Grassmannian variety play a central role. Related codes, namely the affine Grassmann codes, were introduced more recently in Beelen et al. (IEEE Trans Inf Theory 56(7):3166-3176, 2010), while their duals were introduced and studied in Beelen et al. (IEEE Trans Inf Theory 58(6):3843-3855, 2010). In this paper we also classify and count the minimum weight codewords of the dual affine Grassmann codes. Combining the above classification results, we are able to show that the dual of a Grassmann code is generated by its minimum weight codewords. We use these properties to establish that the increase of value of successive generalized Hamming weights of a dual Grassmann code is 1 or 2.
We discuss error floor asympotics and present a method for improving the performance of low-density parity-check (LDPC) codes in the high SNR (error floor) region. The method is based on tanner graph covers that do no...
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We discuss error floor asympotics and present a method for improving the performance of low-density parity-check (LDPC) codes in the high SNR (error floor) region. The method is based on tanner graph covers that do not have trapping sets from the original code. The advantages of the method are that it is universal, as it can be applied to any LDPC code/channel/decoding algorithm and it improves performance at the expense of increasing the code length, without losing the code regularity, without changing the decoding algorithm, and, under certain conditions, without lowering the code rate. The proposed method can be modified to construct convolutional LDPC codes also. The method is illustrated by modifying tanner, MacKay and Margulis codes to improve performance on the binary symmetric channel (BSC) under the Gallager B decoding algorithm. Decoding results on AWGN channel are also presented to illustrate that optimizing codes for one channel/decoding algorithm can lead to performance improvement on other channels.
We explicitly determine an information set for the affine Grassmann codes of an arbitrary level and then use it to describe a systematic encoder for these codes. In the case of affine Grassmann codes of full level, we...
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ISBN:
(纸本)9781467383080
We explicitly determine an information set for the affine Grassmann codes of an arbitrary level and then use it to describe a systematic encoder for these codes. In the case of affine Grassmann codes of full level, we use our explicit information set together with some known results concerning duals of affine Grassmann codes to describe an iterative encoding algorithm and also show that permutation decoding is possible up to a reasonable bound.
Quasi-cyclic codes over finite fields are an important class of linear block codes. A fundamental problem in the theory of these codes is to describe their algebraic structure. In this paper it is shown that every qua...
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Quasi-cyclic codes over finite fields are an important class of linear block codes. A fundamental problem in the theory of these codes is to describe their algebraic structure. In this paper it is shown that every quasi-cyclic code is the subfield code and the trace code of a quasi-cyclic code over an extension field. The latter is defined by a parity check matrix obtained from a spectral analysis of a reduced Grobner basis of the former. Moreover, it is shown that the quasi-cyclic code over the extension field and the one under consideration have the same length, dimension and minimum Hamming distance. Furthermore, we show that under certain conditions it is possible to construct a generator matrix of the quasi-cyclic code over the extension field using similar techniques to construct its parity check matrix. We illustrate that this construction is attainable for some good quasi-cyclic low density parity check codes like the [155, 64, 20] binary tanner code. Copyright (C) 2022 The Authors.
Quasi-cyclic codes over finite fields are an important class of linear block codes. A fundamental problem in the theory of these codes is to describe their algebraic structure. In this paper it is shown that every qua...
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Quasi-cyclic codes over finite fields are an important class of linear block codes. A fundamental problem in the theory of these codes is to describe their algebraic structure. In this paper it is shown that every quasi-cyclic code is the subfield code and the trace code of a quasi-cyclic code over an extension field. The latter is defined by a parity check matrix obtained from a spectral analysis of a reduced Gröbner basis of the former. Moreover, it is shown that the quasi-cyclic code over the extension field and the one under consideration have the same length, dimension and minimum Hamming distance. Furthermore, we show that under certain conditions it is possible to construct a generator matrix of the quasi-cyclic code over the extension field using similar techniques to construct its parity check matrix. We illustrate that this construction is attainable for some good quasi-cyclic low density parity check codes like the [155, 64, 20] binary tanner code.
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